In this note, we consider a single machine scheduling problem with generalized total tardiness objective function. An NP-hardness proof and a pseudo-polynomial time solution algorithm are proposed for a special case of this problem. Moreover, we present a new graphical algorithm for another special case, which corresponds to the classical problem of minimizing the weighted number of tardy jobs on a single machine. The latter algorithm improves the complexity of an existing pseudo-polynomial algorithm by Lawler.